Properties

Label 4010.2.a.i
Level 4010
Weight 2
Character orbit 4010.a
Self dual yes
Analytic conductor 32.020
Analytic rank 1
Dimension 10
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} -\beta_{7} q^{3} + q^{4} + q^{5} + \beta_{7} q^{6} + ( -1 + \beta_{3} + \beta_{7} ) q^{7} - q^{8} + ( \beta_{1} + \beta_{6} ) q^{9} +O(q^{10})\) \( q - q^{2} -\beta_{7} q^{3} + q^{4} + q^{5} + \beta_{7} q^{6} + ( -1 + \beta_{3} + \beta_{7} ) q^{7} - q^{8} + ( \beta_{1} + \beta_{6} ) q^{9} - q^{10} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{11} -\beta_{7} q^{12} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{13} + ( 1 - \beta_{3} - \beta_{7} ) q^{14} -\beta_{7} q^{15} + q^{16} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{17} + ( -\beta_{1} - \beta_{6} ) q^{18} + ( -1 - \beta_{5} - \beta_{7} - \beta_{8} ) q^{19} + q^{20} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{21} + ( 1 + \beta_{6} - \beta_{7} - \beta_{9} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{23} + \beta_{7} q^{24} + q^{25} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{26} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{8} ) q^{27} + ( -1 + \beta_{3} + \beta_{7} ) q^{28} + ( -1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{29} + \beta_{7} q^{30} + ( -2 - \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} - 2 \beta_{6} + \beta_{7} - \beta_{9} ) q^{31} - q^{32} + ( -2 + 2 \beta_{4} + 3 \beta_{7} ) q^{33} + ( -1 - \beta_{1} + \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{34} + ( -1 + \beta_{3} + \beta_{7} ) q^{35} + ( \beta_{1} + \beta_{6} ) q^{36} + ( -1 + \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{37} + ( 1 + \beta_{5} + \beta_{7} + \beta_{8} ) q^{38} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 2 \beta_{7} + \beta_{8} ) q^{39} - q^{40} + ( 1 - \beta_{1} + \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - 2 \beta_{9} ) q^{41} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{6} - \beta_{7} + \beta_{9} ) q^{42} + ( 2 \beta_{1} - \beta_{2} - \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - \beta_{9} ) q^{43} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{44} + ( \beta_{1} + \beta_{6} ) q^{45} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{46} + ( -1 - 2 \beta_{1} + 3 \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - 3 \beta_{8} ) q^{47} -\beta_{7} q^{48} + ( -2 \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{7} + \beta_{8} + \beta_{9} ) q^{49} - q^{50} + ( -2 - 3 \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{6} - \beta_{8} + 2 \beta_{9} ) q^{51} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{52} + ( 4 - 2 \beta_{1} - \beta_{2} - 5 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} - 2 \beta_{8} - \beta_{9} ) q^{53} + ( 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{6} + \beta_{8} ) q^{54} + ( -1 - \beta_{6} + \beta_{7} + \beta_{9} ) q^{55} + ( 1 - \beta_{3} - \beta_{7} ) q^{56} + ( 3 + \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} ) q^{57} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{5} - 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{58} + ( -4 + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} ) q^{59} -\beta_{7} q^{60} + ( -5 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} + \beta_{8} + \beta_{9} ) q^{61} + ( 2 + \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{9} ) q^{62} + ( -1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + \beta_{8} ) q^{63} + q^{64} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{65} + ( 2 - 2 \beta_{4} - 3 \beta_{7} ) q^{66} + ( -2 + \beta_{4} + \beta_{8} + 2 \beta_{9} ) q^{67} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{68} + ( -2 \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} - \beta_{6} - \beta_{8} ) q^{69} + ( 1 - \beta_{3} - \beta_{7} ) q^{70} + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{4} + 2 \beta_{5} + \beta_{8} - \beta_{9} ) q^{71} + ( -\beta_{1} - \beta_{6} ) q^{72} + ( -2 + 2 \beta_{3} + 2 \beta_{4} + 3 \beta_{7} ) q^{73} + ( 1 - \beta_{1} - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{8} ) q^{74} -\beta_{7} q^{75} + ( -1 - \beta_{5} - \beta_{7} - \beta_{8} ) q^{76} + ( 1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{9} ) q^{77} + ( -\beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} + 2 \beta_{7} - \beta_{8} ) q^{78} + ( -3 + 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{7} ) q^{79} + q^{80} + ( -2 + 3 \beta_{1} + \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} ) q^{81} + ( -1 + \beta_{1} - \beta_{3} + \beta_{4} + 2 \beta_{5} + \beta_{7} - 2 \beta_{8} + 2 \beta_{9} ) q^{82} + ( -\beta_{2} + \beta_{4} + \beta_{5} - 2 \beta_{6} + 3 \beta_{7} - 2 \beta_{8} + 3 \beta_{9} ) q^{83} + ( -2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} ) q^{84} + ( 1 + \beta_{1} - \beta_{2} + \beta_{4} - \beta_{5} + \beta_{8} ) q^{85} + ( -2 \beta_{1} + \beta_{2} + \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{7} + \beta_{9} ) q^{86} + ( 5 + \beta_{1} - \beta_{2} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{87} + ( 1 + \beta_{6} - \beta_{7} - \beta_{9} ) q^{88} + ( -3 + \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + 2 \beta_{8} - \beta_{9} ) q^{89} + ( -\beta_{1} - \beta_{6} ) q^{90} + ( -5 + 2 \beta_{2} + \beta_{3} - 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{7} - \beta_{8} + \beta_{9} ) q^{91} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{92} + ( -4 + 3 \beta_{1} + 6 \beta_{3} - 3 \beta_{5} + 2 \beta_{6} + 5 \beta_{7} + 3 \beta_{8} + 2 \beta_{9} ) q^{93} + ( 1 + 2 \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{7} + 3 \beta_{8} ) q^{94} + ( -1 - \beta_{5} - \beta_{7} - \beta_{8} ) q^{95} + \beta_{7} q^{96} + ( 1 + \beta_{2} + \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{6} + 3 \beta_{8} - 2 \beta_{9} ) q^{97} + ( 2 \beta_{2} - \beta_{4} - \beta_{6} + 2 \beta_{7} - \beta_{8} - \beta_{9} ) q^{98} + ( -4 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} - \beta_{7} + 2 \beta_{8} - \beta_{9} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{2} - 4q^{3} + 10q^{4} + 10q^{5} + 4q^{6} - 3q^{7} - 10q^{8} + 6q^{9} + O(q^{10}) \) \( 10q - 10q^{2} - 4q^{3} + 10q^{4} + 10q^{5} + 4q^{6} - 3q^{7} - 10q^{8} + 6q^{9} - 10q^{10} - 11q^{11} - 4q^{12} + 6q^{13} + 3q^{14} - 4q^{15} + 10q^{16} + 9q^{17} - 6q^{18} - 13q^{19} + 10q^{20} - 24q^{21} + 11q^{22} - 3q^{23} + 4q^{24} + 10q^{25} - 6q^{26} - 10q^{27} - 3q^{28} - 4q^{29} + 4q^{30} - 17q^{31} - 10q^{32} - 2q^{33} - 9q^{34} - 3q^{35} + 6q^{36} - 15q^{37} + 13q^{38} - 6q^{39} - 10q^{40} - 11q^{41} + 24q^{42} - 11q^{43} - 11q^{44} + 6q^{45} + 3q^{46} + 3q^{47} - 4q^{48} - 5q^{49} - 10q^{50} - 21q^{51} + 6q^{52} + 25q^{53} + 10q^{54} - 11q^{55} + 3q^{56} + 31q^{57} + 4q^{58} - 46q^{59} - 4q^{60} - 54q^{61} + 17q^{62} - 6q^{63} + 10q^{64} + 6q^{65} + 2q^{66} - 26q^{67} + 9q^{68} - 9q^{69} + 3q^{70} - 16q^{71} - 6q^{72} + 4q^{73} + 15q^{74} - 4q^{75} - 13q^{76} + 11q^{77} + 6q^{78} - 19q^{79} + 10q^{80} - 6q^{81} + 11q^{82} + 19q^{83} - 24q^{84} + 9q^{85} + 11q^{86} + 28q^{87} + 11q^{88} - 30q^{89} - 6q^{90} - 38q^{91} - 3q^{92} - 18q^{93} - 3q^{94} - 13q^{95} + 4q^{96} - 16q^{97} + 5q^{98} - 59q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} - 3 x^{9} - 12 x^{8} + 34 x^{7} + 46 x^{6} - 104 x^{5} - 90 x^{4} + 89 x^{3} + 82 x^{2} + 12 x - 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -129 \nu^{9} + 529 \nu^{8} + 1066 \nu^{7} - 5745 \nu^{6} - 889 \nu^{5} + 16145 \nu^{4} - 1628 \nu^{3} - 12944 \nu^{2} - 352 \nu - 343 \)\()/647\)
\(\beta_{3}\)\(=\)\((\)\( 388 \nu^{9} - 1235 \nu^{8} - 4415 \nu^{7} + 14195 \nu^{6} + 14355 \nu^{5} - 45275 \nu^{4} - 18596 \nu^{3} + 45939 \nu^{2} + 6646 \nu - 5975 \)\()/1941\)
\(\beta_{4}\)\(=\)\((\)\( -623 \nu^{9} + 1978 \nu^{8} + 7024 \nu^{7} - 21847 \nu^{6} - 24225 \nu^{5} + 63547 \nu^{4} + 45307 \nu^{3} - 52647 \nu^{2} - 39236 \nu - 1817 \)\()/1941\)
\(\beta_{5}\)\(=\)\((\)\( -304 \nu^{9} + 1041 \nu^{8} + 3119 \nu^{7} - 11402 \nu^{6} - 8239 \nu^{5} + 32505 \nu^{4} + 11215 \nu^{3} - 26075 \nu^{2} - 11337 \nu - 61 \)\()/647\)
\(\beta_{6}\)\(=\)\((\)\( -913 \nu^{9} + 2771 \nu^{8} + 10574 \nu^{7} - 31166 \nu^{6} - 36405 \nu^{5} + 94355 \nu^{4} + 57125 \nu^{3} - 83391 \nu^{2} - 41542 \nu + 998 \)\()/1941\)
\(\beta_{7}\)\(=\)\((\)\( 983 \nu^{9} - 3364 \nu^{8} - 10360 \nu^{7} + 37699 \nu^{6} + 29640 \nu^{5} - 113839 \nu^{4} - 45052 \nu^{3} + 103395 \nu^{2} + 46583 \nu - 205 \)\()/1941\)
\(\beta_{8}\)\(=\)\((\)\( -1162 \nu^{9} + 4409 \nu^{8} + 10811 \nu^{7} - 48665 \nu^{6} - 19689 \nu^{5} + 142145 \nu^{4} + 10769 \nu^{3} - 125544 \nu^{2} - 20404 \nu + 7799 \)\()/1941\)
\(\beta_{9}\)\(=\)\((\)\( -1720 \nu^{9} + 5975 \nu^{8} + 18311 \nu^{7} - 68189 \nu^{6} - 52830 \nu^{5} + 213110 \nu^{4} + 74912 \nu^{3} - 201486 \nu^{2} - 72844 \nu + 4916 \)\()/1941\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - \beta_{5} + \beta_{3} + \beta_{2} + \beta_{1} + 3\)
\(\nu^{3}\)\(=\)\(\beta_{8} + \beta_{6} - \beta_{5} + 2 \beta_{3} - \beta_{2} + 7 \beta_{1} + 1\)
\(\nu^{4}\)\(=\)\(-\beta_{9} + 3 \beta_{8} - \beta_{7} + 10 \beta_{6} - 12 \beta_{5} + \beta_{4} + 10 \beta_{3} + 6 \beta_{2} + 11 \beta_{1} + 19\)
\(\nu^{5}\)\(=\)\(13 \beta_{8} + 3 \beta_{7} + 15 \beta_{6} - 15 \beta_{5} + \beta_{4} + 24 \beta_{3} - 9 \beta_{2} + 56 \beta_{1} + 9\)
\(\nu^{6}\)\(=\)\(-10 \beta_{9} + 41 \beta_{8} - 2 \beta_{7} + 93 \beta_{6} - 114 \beta_{5} + 16 \beta_{4} + 97 \beta_{3} + 37 \beta_{2} + 108 \beta_{1} + 135\)
\(\nu^{7}\)\(=\)\(\beta_{9} + 142 \beta_{8} + 52 \beta_{7} + 171 \beta_{6} - 180 \beta_{5} + 29 \beta_{4} + 250 \beta_{3} - 74 \beta_{2} + 478 \beta_{1} + 85\)
\(\nu^{8}\)\(=\)\(-74 \beta_{9} + 449 \beta_{8} + 70 \beta_{7} + 848 \beta_{6} - 1042 \beta_{5} + 195 \beta_{4} + 927 \beta_{3} + 229 \beta_{2} + 1046 \beta_{1} + 1014\)
\(\nu^{9}\)\(=\)\(25 \beta_{9} + 1462 \beta_{8} + 660 \beta_{7} + 1784 \beta_{6} - 1969 \beta_{5} + 445 \beta_{4} + 2508 \beta_{3} - 600 \beta_{2} + 4229 \beta_{1} + 848\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.08731
1.47143
−1.54628
0.0585304
−0.557507
1.96171
−2.65028
−1.06458
−0.309245
2.54891
−1.00000 −3.13000 1.00000 1.00000 3.13000 2.49851 −1.00000 6.79692 −1.00000
1.2 −1.00000 −2.70072 1.00000 1.00000 2.70072 −1.21166 −1.00000 4.29386 −1.00000
1.3 −1.00000 −1.74362 1.00000 1.00000 1.74362 2.79885 −1.00000 0.0402221 −1.00000
1.4 −1.00000 −1.47624 1.00000 1.00000 1.47624 −2.32277 −1.00000 −0.820714 −1.00000
1.5 −1.00000 −1.25862 1.00000 1.00000 1.25862 1.88678 −1.00000 −1.41588 −1.00000
1.6 −1.00000 0.223737 1.00000 1.00000 −0.223737 −0.465545 −1.00000 −2.94994 −1.00000
1.7 −1.00000 0.675556 1.00000 1.00000 −0.675556 −1.96617 −1.00000 −2.54362 −1.00000
1.8 −1.00000 0.696154 1.00000 1.00000 −0.696154 2.65990 −1.00000 −2.51537 −1.00000
1.9 −1.00000 2.30797 1.00000 1.00000 −2.30797 −5.12569 −1.00000 2.32671 −1.00000
1.10 −1.00000 2.40579 1.00000 1.00000 −2.40579 −1.75221 −1.00000 2.78781 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4010.2.a.i 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4010.2.a.i 10 1.a even 1 1 trivial

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(401\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{10} + \cdots\)
\(T_{7}^{10} + \cdots\)
\(T_{11}^{10} + \cdots\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{10} \)
$3$ \( 1 + 4 T + 20 T^{2} + 58 T^{3} + 182 T^{4} + 434 T^{5} + 1076 T^{6} + 2235 T^{7} + 4753 T^{8} + 8719 T^{9} + 16208 T^{10} + 26157 T^{11} + 42777 T^{12} + 60345 T^{13} + 87156 T^{14} + 105462 T^{15} + 132678 T^{16} + 126846 T^{17} + 131220 T^{18} + 78732 T^{19} + 59049 T^{20} \)
$5$ \( ( 1 - T )^{10} \)
$7$ \( 1 + 3 T + 42 T^{2} + 129 T^{3} + 894 T^{4} + 2832 T^{5} + 12609 T^{6} + 39725 T^{7} + 130193 T^{8} + 387735 T^{9} + 1033046 T^{10} + 2714145 T^{11} + 6379457 T^{12} + 13625675 T^{13} + 30274209 T^{14} + 47597424 T^{15} + 105178206 T^{16} + 106237047 T^{17} + 242121642 T^{18} + 121060821 T^{19} + 282475249 T^{20} \)
$11$ \( 1 + 11 T + 103 T^{2} + 616 T^{3} + 3268 T^{4} + 13096 T^{5} + 47861 T^{6} + 133699 T^{7} + 360987 T^{8} + 744096 T^{9} + 2350040 T^{10} + 8185056 T^{11} + 43679427 T^{12} + 177953369 T^{13} + 700732901 T^{14} + 2109123896 T^{15} + 5789461348 T^{16} + 12004097336 T^{17} + 22078964743 T^{18} + 25937424601 T^{19} + 25937424601 T^{20} \)
$13$ \( 1 - 6 T + 81 T^{2} - 287 T^{3} + 2372 T^{4} - 4060 T^{5} + 34023 T^{6} + 16767 T^{7} + 283911 T^{8} + 1210928 T^{9} + 2420282 T^{10} + 15742064 T^{11} + 47980959 T^{12} + 36837099 T^{13} + 971730903 T^{14} - 1507449580 T^{15} + 11449190948 T^{16} - 18008824379 T^{17} + 66074188401 T^{18} - 63626996238 T^{19} + 137858491849 T^{20} \)
$17$ \( 1 - 9 T + 152 T^{2} - 1037 T^{3} + 10141 T^{4} - 56380 T^{5} + 409608 T^{6} - 1923285 T^{7} + 11325598 T^{8} - 45513113 T^{9} + 225415358 T^{10} - 773722921 T^{11} + 3273097822 T^{12} - 9449099205 T^{13} + 34210869768 T^{14} - 80051537660 T^{15} + 244779087229 T^{16} - 425521203901 T^{17} + 1060315131032 T^{18} - 1067290888473 T^{19} + 2015993900449 T^{20} \)
$19$ \( 1 + 13 T + 192 T^{2} + 1664 T^{3} + 14659 T^{4} + 97713 T^{5} + 648691 T^{6} + 3562244 T^{7} + 19446002 T^{8} + 91270928 T^{9} + 426588208 T^{10} + 1734147632 T^{11} + 7020006722 T^{12} + 24433431596 T^{13} + 84538059811 T^{14} + 241947061587 T^{15} + 689645569579 T^{16} + 1487402573696 T^{17} + 3260844103872 T^{18} + 4194940071127 T^{19} + 6131066257801 T^{20} \)
$23$ \( 1 + 3 T + 187 T^{2} + 425 T^{3} + 16093 T^{4} + 27993 T^{5} + 855348 T^{6} + 1169460 T^{7} + 31480234 T^{8} + 35291365 T^{9} + 842951618 T^{10} + 811701395 T^{11} + 16653043786 T^{12} + 14228819820 T^{13} + 239361439668 T^{14} + 180172549599 T^{15} + 2382341561677 T^{16} + 1447050814975 T^{17} + 14644154247547 T^{18} + 5403457984389 T^{19} + 41426511213649 T^{20} \)
$29$ \( 1 + 4 T + 129 T^{2} + 570 T^{3} + 7726 T^{4} + 35616 T^{5} + 303517 T^{6} + 1267961 T^{7} + 9743651 T^{8} + 32792993 T^{9} + 289804684 T^{10} + 950996797 T^{11} + 8194410491 T^{12} + 30924300829 T^{13} + 214671807277 T^{14} + 730525082784 T^{15} + 4595604978046 T^{16} + 9832429496130 T^{17} + 64531787271969 T^{18} + 58028583903476 T^{19} + 420707233300201 T^{20} \)
$31$ \( 1 + 17 T + 216 T^{2} + 2088 T^{3} + 18749 T^{4} + 145342 T^{5} + 1063788 T^{6} + 7150461 T^{7} + 46404448 T^{8} + 277307962 T^{9} + 1596171690 T^{10} + 8596546822 T^{11} + 44594674528 T^{12} + 213019383651 T^{13} + 982430557548 T^{14} + 4161018064642 T^{15} + 16639806515069 T^{16} + 57446338263768 T^{17} + 184224464087256 T^{18} + 449473576731407 T^{19} + 819628286980801 T^{20} \)
$37$ \( 1 + 15 T + 318 T^{2} + 3739 T^{3} + 45945 T^{4} + 441510 T^{5} + 4052846 T^{6} + 32751775 T^{7} + 244564024 T^{8} + 1686062901 T^{9} + 10602280982 T^{10} + 62384327337 T^{11} + 334808148856 T^{12} + 1658975659075 T^{13} + 7595685912206 T^{14} + 30616050455070 T^{15} + 117882299861505 T^{16} + 354950288600287 T^{17} + 1116968466346878 T^{18} + 1949426096926155 T^{19} + 4808584372417849 T^{20} \)
$41$ \( 1 + 11 T + 238 T^{2} + 2000 T^{3} + 28011 T^{4} + 207360 T^{5} + 2280094 T^{6} + 14946485 T^{7} + 137235898 T^{8} + 799593928 T^{9} + 6373618252 T^{10} + 32783351048 T^{11} + 230693544538 T^{12} + 1030126692685 T^{13} + 6443000701534 T^{14} + 24023941839360 T^{15} + 133055169894651 T^{16} + 389508547762000 T^{17} + 1900412204530798 T^{18} + 3601201278333571 T^{19} + 13422659310152401 T^{20} \)
$43$ \( 1 + 11 T + 197 T^{2} + 1901 T^{3} + 22094 T^{4} + 191017 T^{5} + 1740213 T^{6} + 13397158 T^{7} + 106178861 T^{8} + 725466695 T^{9} + 5107982132 T^{10} + 31195067885 T^{11} + 196324713989 T^{12} + 1065167841106 T^{13} + 5949441944613 T^{14} + 28081111756531 T^{15} + 139664195204606 T^{16} + 516727179714407 T^{17} + 2302575454687397 T^{18} + 5528518731305273 T^{19} + 21611482313284249 T^{20} \)
$47$ \( 1 - 3 T + 194 T^{2} - 323 T^{3} + 17109 T^{4} - 940 T^{5} + 811722 T^{6} + 2405797 T^{7} + 19065686 T^{8} + 231609061 T^{9} + 317343936 T^{10} + 10885625867 T^{11} + 42116100374 T^{12} + 249777061931 T^{13} + 3960944420682 T^{14} - 215584306580 T^{15} + 184421595063861 T^{16} - 163639267909549 T^{17} + 4619389612381634 T^{18} - 3357391419308301 T^{19} + 52599132235830049 T^{20} \)
$53$ \( 1 - 25 T + 427 T^{2} - 5582 T^{3} + 64602 T^{4} - 665382 T^{5} + 6336438 T^{6} - 56080783 T^{7} + 467737103 T^{8} - 3708855556 T^{9} + 27786225496 T^{10} - 196569344468 T^{11} + 1313873522327 T^{12} - 8349138730691 T^{13} + 49997543646678 T^{14} - 278259753523326 T^{15} + 1431862057655658 T^{16} - 6557237582570134 T^{17} + 26584887805651147 T^{18} - 82494089795053325 T^{19} + 174887470365513049 T^{20} \)
$59$ \( 1 + 46 T + 1370 T^{2} + 29248 T^{3} + 507730 T^{4} + 7358451 T^{5} + 92806871 T^{6} + 1028827578 T^{7} + 10218323913 T^{8} + 91125712475 T^{9} + 736477435704 T^{10} + 5376417036025 T^{11} + 35569985541153 T^{12} + 211299579142062 T^{13} + 1124574359187431 T^{14} + 5260735422900849 T^{15} + 21416322345544930 T^{16} + 72788078627986112 T^{17} + 201157699517919770 T^{18} + 398497807658127194 T^{19} + 511116753300641401 T^{20} \)
$61$ \( 1 + 54 T + 1716 T^{2} + 39282 T^{3} + 715319 T^{4} + 10864488 T^{5} + 141988979 T^{6} + 1627514861 T^{7} + 16582577610 T^{8} + 151415379327 T^{9} + 1245270340302 T^{10} + 9236338138947 T^{11} + 61703771286810 T^{12} + 369414950664641 T^{13} + 1965956826986339 T^{14} + 9176106377058888 T^{15} + 36853502667536159 T^{16} + 123453224084576922 T^{17} + 328969749103334196 T^{18} + 631483889013043614 T^{19} + 713342911662882601 T^{20} \)
$67$ \( 1 + 26 T + 703 T^{2} + 12197 T^{3} + 199474 T^{4} + 2642918 T^{5} + 32795787 T^{6} + 354040609 T^{7} + 3605991281 T^{8} + 32798357058 T^{9} + 283115362804 T^{10} + 2197489922886 T^{11} + 16187294860409 T^{12} + 106482315684667 T^{13} + 660871872127227 T^{14} + 3568269947542226 T^{15} + 18044095324779106 T^{16} + 73922499450124631 T^{17} + 285465577322318623 T^{18} + 707369894303668622 T^{19} + 1822837804551761449 T^{20} \)
$71$ \( 1 + 16 T + 658 T^{2} + 7807 T^{3} + 181148 T^{4} + 1676154 T^{5} + 28845062 T^{6} + 216422887 T^{7} + 3107133973 T^{8} + 19751330864 T^{9} + 250233628330 T^{10} + 1402344491344 T^{11} + 15663062357893 T^{12} + 77460131909057 T^{13} + 733001513969222 T^{14} + 3024166243596054 T^{15} + 23205110231721308 T^{16} + 71005603076558537 T^{17} + 424905823559710738 T^{18} + 733576011495184496 T^{19} + 3255243551009881201 T^{20} \)
$73$ \( 1 - 4 T + 516 T^{2} - 1626 T^{3} + 129718 T^{4} - 334152 T^{5} + 20927890 T^{6} - 45246887 T^{7} + 2391701595 T^{8} - 4417520027 T^{9} + 201898642960 T^{10} - 322478961971 T^{11} + 12745377799755 T^{12} - 17601808240079 T^{13} + 594315263841490 T^{14} - 692721018944136 T^{15} + 19630773165756502 T^{16} - 17963069992051722 T^{17} + 416133407417345796 T^{18} - 235486346833071652 T^{19} + 4297625829703557649 T^{20} \)
$79$ \( 1 + 19 T + 701 T^{2} + 10669 T^{3} + 223488 T^{4} + 2834077 T^{5} + 43314051 T^{6} + 466164732 T^{7} + 5686462673 T^{8} + 52208581403 T^{9} + 529742938534 T^{10} + 4124477930837 T^{11} + 35489213542193 T^{12} + 229837393300548 T^{13} + 1687085794888131 T^{14} + 8720614768108723 T^{15} + 54327129259477248 T^{16} + 204886504973330371 T^{17} + 1063493275744499261 T^{18} + 2277180323669748061 T^{19} + 9468276082626847201 T^{20} \)
$83$ \( 1 - 19 T + 687 T^{2} - 11547 T^{3} + 227142 T^{4} - 3257653 T^{5} + 46742113 T^{6} - 563300086 T^{7} + 6536553969 T^{8} - 66156590137 T^{9} + 643472849516 T^{10} - 5490996981371 T^{11} + 45030320292441 T^{12} - 322087666273682 T^{13} + 2218302202972273 T^{14} - 12832027567790879 T^{15} + 74261890287781398 T^{16} - 313339980777222969 T^{17} + 1547324763479521167 T^{18} - 3551864850083267657 T^{19} + 15516041187205853449 T^{20} \)
$89$ \( 1 + 30 T + 1011 T^{2} + 20699 T^{3} + 418614 T^{4} + 6544860 T^{5} + 98981835 T^{6} + 1250450097 T^{7} + 15230285299 T^{8} + 159866982622 T^{9} + 1617442689892 T^{10} + 14228161453358 T^{11} + 120639089853379 T^{12} + 881528554431993 T^{13} + 6210342146192235 T^{14} + 36546887325382140 T^{15} + 208043326134348054 T^{16} + 915544401002554771 T^{17} + 3979891282564803891 T^{18} + 10510692111224556270 T^{19} + 31181719929966183601 T^{20} \)
$97$ \( 1 + 16 T + 640 T^{2} + 6946 T^{3} + 179074 T^{4} + 1534105 T^{5} + 33304185 T^{6} + 244454186 T^{7} + 4690150175 T^{8} + 30216640161 T^{9} + 513900985612 T^{10} + 2931014095617 T^{11} + 44129622996575 T^{12} + 223106735299178 T^{13} + 2948395552340985 T^{14} + 13173881624964985 T^{15} + 149163628810655746 T^{16} + 561224883984972898 T^{17} + 5015957500401255040 T^{18} + 12163696938473043472 T^{19} + 73742412689492826049 T^{20} \)
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